A new algorithm for the maximum likelihood estimation of graphical log-linear models
type: Item
instance of: scholarly article
title: A new algorithm for the maximum likelihood estimation of graphical log-linear models
zbMATH Open document ID: 0923.62057
author: Tamás Rudas
published in: Computational Statistics
publication date: 1999-09-14T00:00:00Z
review text: A \(p\)-dimensional contingency table \(\Omega\) formed by the variables \(V=\{V_{1},\ldots, V_{p}\}\) is the Cartesian product of the sets of indices of the variables in \(V\). A log-linear model \(LL(\Gamma)\) is defined by a class \(\Gamma\) of subsets of \(V\) and it is assumed that \(P(\omega)=\prod_{\gamma\in\Gamma}g_{\gamma}(\omega_{\gamma})\), where \(g_{\gamma}:\Omega_{\gamma}\to R,\;\omega_{\gamma}\in\Omega_{\gamma}\), is a marginal cell of the marginal table defined by the subset of variables \(V_{\gamma},\;\gamma\in\Gamma\). A class \(\Gamma\) which does not contain comparable subsets is called a generating class. A generating class \(\Gamma\) or log-linear model \(LL(\Gamma)\) generated by \(\Gamma\) is called graphical if there are no subsets \(W\subset V\) for which \(| W|\geq 3\) and every \(W'\subset W\) with \(| W'|=| W|-1\) is a generator. Let the first order interaction graph of \(\Gamma\) be denoted by \(G(\Gamma)\). The author proves the following result: Let \(\Gamma\) be a graphical generating class. A distribution \(P\) belongs to \(LL(\Gamma)\) if and only if for every maximal complete subgraph \(W\) of \(\bar G(\Gamma)\) the joint conditional distribution of the variables in \(W\), given all other variables, is independent on \(P\). Then an algorithm to compute maximum likelihood estimates under graphical log-linear models is proposed and a proof of convergence is given.
Mathematics Subject Classification ID: 62H17
Mathematics Subject Classification ID: 62J12
Mathematics Subject Classification ID: 05C90
Mathematics Subject Classification ID: 62A09
zbMATH DE Number: 1336643
zbMATH Keywords: contingency tables
zbMATH Keywords: log-linear models
zbMATH Keywords: graphical log-linear models
zbMATH Keywords: maximum likelihood estimation
reviewed by: Aleksandr D. Borisenko
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1336643